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by rmazzawi » Thu Sep 15, 2011 10:05 am
The roots of the quadratic equation x2 - ax + b = 0 are both integers. If 'b' is a prime number and 'a' is a positive integer, then which of the following could be the value of the sum of the roots of the quadratic equation?
24
29
57
40
92
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by cans » Thu Sep 15, 2011 10:14 am
sum of the roots = a.
product=b.
if b is prime number, one root=1 and other=b
thus a = b+1.
24=23+1 possible.
29=28+1 nopes
57=56+1 nopes
40=39+1 nopes
92=91+1 nopes
IMO A
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by knight247 » Thu Sep 15, 2011 10:18 am
In a quadratic equation of the form ax2+bx+c the sum of roots=-b/a and product of roots=c/a

using this formula on x2 - ax + b = 0 we have
sum of roots=-(-a)=a ....+ve integer
product of roots=b ....which is a prime number meaning that its only factors could be 1 and itself. So the two roots are 1 and some other prime number. So sum of roots=1+some prime number.We need to find which of the answer choices would give a prime number when 1 is subtracted from them.
(A)24-1=23 YES
(B)29-1=28 NO
(C)57-1=56 NO
(D)40-1=39 NO
(E)92-1=91 NO

Hence A
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by sl750 » Thu Sep 15, 2011 10:20 am
Ok, I screwed it up

Sum of roots x1+x2 = a

The factors for a prime number are the number itself and 1.
We need a number for a, for which a-1 is a prime number


For a=24 b=23; x^2-24x+23=0 Can be expressed as the factors of the prime number 23 (x-23)(x-1)
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by tpr-becky » Thu Sep 15, 2011 11:51 am
To factor the quadratic we know that b = the product of the roots and a = the sum of the roots. The signs are controlled by the signs in the equation - if the last sum is positive then the roots are either both positive or both negative, add that to the negative in front of the second term means that each of the roots are positive.

The fact that b = the product of the roots and that b is prime means that b is a multiple of a prime number and 1. thus the sum of the roots is going to be a prime plus one.

That means to subtract 1 from each of the answers and figure out which result is prime. Only A creates this result.
Becky
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The Princeton Review
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